36 ON THE EFFECT OF THE INTEKNAL FRICTION OF FLUIDS It turns out that K is a complicated function of m and aS"1, and the algebraical expressions for the quantities which answer to k and k' in Art. 20 would be more complicated still, because v (i _}. V- 1) would have to be substituted for m in (60) and (59), and then K reduced to the form — k4- V— Ik'. To obtain numerical results from these formulae, it would be best to substitute the numerical values of a, b, and v in (60) and (59), and perform the reduction of K in figures. 22. If the distance of the envelope from the surface of the sphere be at all considerable, the exponential ev(5~a)} which arises from ew(5~a), will have so large a numerical value that we may neglect the terms in the numerator and denominator of the fraction in the expression for K which contain e~v&-<*)9 as well as the term in the denominator which is free from exponentials, in comparison with the terms which contain ev^"a\ Thus, if 5- a be two inches, rone second, and \/// = -116, we have e"(5-^=2424000000, nearly; and if 5 — a be only an inch or half an inch, we have still the square or fourth root of the above quantity, that is, about 49234 or 222, for the value of that exponential. Hence, in practical cases, the above simplification may be made, which will cause the exponentials to disappear from the expression for K. We thus get 8& (mV + 3ma + 3) (m*V - 3mb + 3) 2mV b (m*V - 3mb + 3) - a (mV + 3ma + 3) ' ' ' (^ '* if we assume 3w + 3 4- (2*V + 3i/a) V^l = A' (cos a + V^T sin a), - 3j# + 3 + (2iW - 3^6) V^l = Bf (cos ft 4- V^I sin /?), l)Bf cos ft — a A' cos a = (7 cos 7, bB' sin ft — a A' sin a = G' sin 7, get from (61) ^=1+ '--cosa + -f ^/::isina - we whence (62V SbA'B' ' .....